Multiplying and Dividing Undo Each Other

One array, read two ways

Here are 12 squares in 3 rows, with 4 squares in each row. Read as a multiplication, 3 rows of 4 make 12, so 3 × 4 = 12.

Read as a division, the same picture shares 12 squares equally into 3 rows, and each row gets 4. So 12 ÷ 3 = 4. The array has not changed: only the question has.

12 squares shared into 3 rows put 4 in each row: 12 ÷ 3 = 4.

Split it by columns

Now look at the columns instead of the rows. There are 4 columns with 3 squares in each, so 12 shared into 4 columns puts 3 in each column: 12 ÷ 4 = 3. The columns also give the multiplication 4 × 3 = 12.

So one picture holds four facts: 3 × 4 = 12, 4 × 3 = 12, 12 ÷ 3 = 4 and 12 ÷ 4 = 3. The four facts use the same three numbers, and together they are called a fact family.

12 squares shared into 4 columns put 3 in each column: 12 ÷ 4 = 3.

One builds it, the other takes it apart

Start with 4 and multiply by 3, and you reach 12. Now divide 12 by 3, and you are back at 4. Multiplying by 3 builds the 12 out of 3 equal groups, and dividing by 3 takes it back apart into those groups. Each one undoes the other.

Deal 12 counters out to 3 friends, one to each friend in turn. Every round gives each friend one more, and after 4 rounds the pile is gone, so each friend has 4: 12 ÷ 3 = 4. Put the 3 shares of 4 back together and they make 3 × 4 = 12, the pile you started with.

Dealing 12 counters to 3 friends takes 4 full rounds, so each friend gets 4. The 3 shares of 4 make 12 again.

Using one to find the other

Because they undo each other, every division has a multiplication that checks it. 12 ÷ 3 = 4 is right because 3 × 4 = 12. To work out 12 ÷ 3, ask "3 times what makes 12?"

The same idea finds a missing number. Some boxes hold 5 pens each and there are 30 pens in all, so the number of boxes times 5 is 30. Dividing undoes the multiplying: 30 ÷ 5 = 6 boxes, and 6 × 5 = 30 checks it. The usual mistake is to multiply the two numbers you are given, 30 × 5, but 30 is already the total.